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Atomic force microscopy’s path to atomicresolution

Franz J. Giessibl

We review the progress in the spatial resolution of atomic force microscopy(AFM) in vacuum. After an introduction of the basic principle and a conceptualcomparison to scanning tunneling microscopy, the main challenges of AFMand the solutions that have evolved in the first twenty years of its existence areoutlined. Some crucial steps along the AFM’s path towards higher resolutionare discussed, followed by an outlook on current and future applications.

Review Feature, submitted to Materials Today Feb 10 2005, 2nd revised ver-

sion

Experimentalphysik VI, EKM,Institute of Physics, Augsburg University,86135 Augsburg, Germany,Email: franz.giessibl@physik.uni-augsburg.de

Atomic force microscopy, invented1 and also introduced2 in 1985/86, can be viewed

as a mechanical profiling technique that generates three-dimensional maps of surfaces by

scanning a sharp probe attached to a cantilever over a surface. The forces that act between

the tip of the cantilever and the sample are used to control the vertical distance. AFM’s

potential to reach atomic resolution was foreseen in the original scientific publication,2 but

for a long time, the spatial resolution of AFM was inferior to the resolution capability of

its parent, scanning tunneling microscopy (STM). The resolution limits of STM and AFM

are given by the structural properties of the atomic wavefunctions of the probe tip and

the sample. STM is sensitive to the most loosely bonded electrons with an energy at the

Fermi level while AFM responds to all electrons, including core electrons. Because the

1

electrons at the Fermi level are spatially less confined thancore electrons that contribute

to AFM images, in theory AFM should be able to achieve even greater spatial resolution

than STM. Today, experimental evidence emerges where in simultaneous AFM/STM stud-

ies, AFM images reveal even finer structural details than simultaneously recorded STM

images. The experimental advances that made high-resolution AFM possible started with

the introduction of frequency modulation AFM (FM-AFM), whe re the cantilever oscillates

at a fixed amplitude and the frequency is used as a feedback signal. Early implementa-

tions of FM-AFM utilized silicon cantilevers with a typical spring constant of 10 N/m that

oscillate with an amplitude on the order of 10 nanometers. The spatial resolution could be

increased by the introduction of quartz cantilevers with a stiffness on the order of 1 kN/m,

allowing the use of sub-nm amplitudes. The direct evaluation of higher harmonics in

the cantilever motion has enabled a further increase in spatial resolution. Because AFM

can image insulators as well as conductors, it is now a powerful complement to STM for

atomically resolved surface studies. Immediate applications of high resolution AFM have

been demonstrated in vacuum studies relating to materials science, surface physics and

-chemistry. Some of the techniques developed for ultrahigh-vacuum AFM may be appli-

cable to increase AFM resolution in ambient or liquid environments, such as required for

studying biological or technological specimens.

1 Introduction

In terms of the operating principle, atomic force microscopy (AFM)1, 2 can be viewed as an

extension of the toddlers way of ‘grasping’ the world by touching and feeling as indicated in

Figure 1 of Binnig and Rohrer’s article ‘In touch with atoms’,3 where a finger profiles an atomic

surface. Likewise, one could argue that stylus profilometryis a predecessor of AFM. However,

AFM and stylus profilometry have as much in common as a candle and a laser. Both of the

2

latter generate light, and even though candles are masterpieces of engineering,4 the laser is a

much more advanced technological device requiring a detailed knowledge of modern quantum

mechanics.5 While stylus profilometry is an extension of human capabilities that have been

known for ages and works by classical mechanics, AFM requires a detailed understanding of

the physics of chemical bonding forces and the technological prowess to measure forces that

are several orders of magnitude smaller than the forces acting in profilometry. Only the spectac-

ular spatial resolution of scanning tunneling microscopy (STM) could trigger the hope that the

force acting between any STM tip and sample might lead to atomic force microscopy capable

of true atomic resolution. The STM, established in 1981, is the first instrument that has allowed

to image surfaces with atomic resolution in real space.6, 7 The atomic imaging of the 7×7 re-

construction of Si (111) by STM in 19838 has later helped to solve one of the most intriguing

problems of surface science at that time and establish the dimer-adatom-stacking fault model by

Takayanagi et al.9 The capability of atomic resolution by STM provided immediate evidence

for the enormous value of this instrument as a tool for surface scientists. STM can only be used

on conductive surfaces. Given that many surfaces of technological interest are conducting or

at least semiconducting, this may not seem to be a severe shortcoming. One might think that

an STM should be capable of mapping the surface of a metallic surface at ambient conditions.

However, this is not feasible, because the pervasive layer of oxides and other contaminants

occurring at ambient conditions prevents stable tunnelingconditions. Electrical conductivity

is a necessary, but not a sufficient condition for a surface tobe imaged by STM with atomic

resolution, because surfaces need to be extremely clean on an atomic level. Except for a few

extremely inert surfaces such as graphite, atomic resolution is only possible in an ultra-high

vacuum with a pressure on the order of 10−8 Pa and special surface preparation. The invention

of the AFM by Binnig1 and its introduction by Binnig, Quate and Gerber2 opened the possibility

of obtaining true atomic resolution on conductorsand insulators. Indeed, it took only a short

3

time after the AFM’s invention before apparently atomic resolution on conductors10 and insu-

lators11–13 was obtained. While these early results reproduced the periodic lattice spacings of

the samples that were studied, single defects or step edges were not observed. Also, the forces

that acted between tip and sample were often orders of magnitudes larger than the forces that

a tip with a single front atom was expected to be able to sustain. Therefore, it was commonly

assumed thatmanytip atoms interacted with the surface at the same time in these early experi-

ments. The difference betweenapparentandtrue atomic resolution of a tip with many atomic

contacts can be illustrated by a macroscopic example: When profiling an egg crate with a single

egg, its trajectory would represent the overall periodicity of the crate as well as a dented hump

or a hole. However, when profiling one egg crate with another egg crate, again its periodicity

would be retained, but holes or dented humps would pass undetected. A similar effect can occur

when an AFM tip probes a surface. As long as single defects, steps or other singularities are

not observed, a clear proof fortrue atomic resolution is not established. Even though atomic

resolution was hardly ever achieved in initial AFM experiments, the technique was readily ac-

cepted and found many technological and scientific applications. The installed base of atomic

force microscopes rapidly outnumbered their STM counterparts. A recent survey14 about the

ten most highly cited publications ofPhysical Review Lettersranks the original AFM publica-

tion2 as number four (4251 citations as of Mar 11 2005 [ISI]) – in good company with other

breakthroughs in theoretical and experimental physics that have shaped our scientific life. Most

of these citations refer to AFM where the spatial resolutionis ‘only’ in the nanometer-range,

but the large number proves the vast applications of AFM. In spite of the rapid growth of AFM

applications, matching and even exceeding the spatial resolution of STM, its parent, had to wait

for new experimental developments.

4

2 Challenges of atomic resolution AFM

The technological foundations for the feasibility of STM with atomic resolution (theory of elec-

tron tunneling, mechanical actuation with pico-meter precision, vacuum technology, surface-

and tip preparation, vibration isolation, ...) were probably available a few decades before 1981,

but it took the bold approach by Gerd Binnig, Heinrich Rohrer, Christoph Gerber and Edi

Weibel to pursue atomic resolution in real space. Binnig andRohrer were rewarded with the

Nobel Prize in Physics in 1986 (together with Ernst Ruska, the inventor of Electron Microscopy)

and the AFM principle was published in the same year. The challenges of AFM with true atomic

resolution are even more daunting than the hurdles that troubled STM. To start our discussion

of the special AFM challenges, we first look at the physics behind STM. Figure 1 (a) shows a

schematic view of a sharp tip for STM or AFM close to a crystalline sample and Fig. 1 (b) is

a plot of the tunneling current and forces between tip and sample. When tip and sample are

conductive and a bias voltage is applied between them, a tunneling current can flow. The red

curve in Fig. 1 (b) shows the distance dependence of the tunneling currentIt. The exponential

decay ofIt with distance at a rate of approximately one order of magnitude per 100 pm distance

increase is the key physical characteristic that makes atomic resolution STM possible. Because

of its strong decay rate, the tunneling current is spatiallyconfined to the front atom of the tip

and flows mainly to the sample atom next to it (indicated by redcircles in Fig. 1 (a)). A second

helpful property of the tunneling current is its monotonic distance dependence. It is easy to

build a feedback mechanism that keeps the tip at a constant distance: if the actual tunneling

current is larger than the setpoint, the feedback needs to withdraw the tip and vice versa. The

tip sample forceFts, in contrast, does not share the helpful key characteristics of the tunneling

current. First,Fts is composed of long-range background forces depicted in light-blue in Fig. 1

(b) and originating from the atoms colored light-blue in Fig. 1 (a) and a short-range component

5

depicted in blue in Fig. 1 (b) and confined to the atoms printedin blue in Fig. 1 (a). Because the

short-range force is not monotonic, it is difficult to designa feedback loop that controls distance

by utilizing the force. A central task to perfect AFM is therefore the isolation the front atom’s

force contribution and the creation of a linear feedback signal from it.

Even if it was possible to isolate the short-range force, a more basic problem needs to be

solved first: how to measure small forces. For example, commonly known force meters such

as precise scales are delicate and expensive instruments and even top models rarely exceed

a mass resolution of 100µg, corresponding to a force resolution of 1µN. In addition, high-

precision scales take about one second to acquire a weight measurement so the bandwidth is

only 1 Hz. The force meters in AFM, in contrast, require a force resolution of at least a nano-

Newton at a typical bandwidth of 1 kHz. Most force meters determine the deflectionq′ of a

spring with given spring constantk that is subject to a forceF with F = q′/k. Measuring

small spring deflections is subject to thermal drift and other noise factors, resulting in a finite

deflection measurement accuracyδq′. The force resolution is thus given byδF = δq′/k, and

soft cantilevers provide less noise in the force measurement. In contact-mode AFM, where the

tip feels small repulsive forces from the sample surface, the cantilever should be softer than

the bonds between surface atoms (estimated at≈ 10 N/m), otherwise the sample deforms more

than the cantilever.15 Because of noise and stability considerations, spring constants below

1 N/m or so have been chosen for AFM in contact mode. However, atomic forces are usually

attractive in the distance regime that is best suited for atomic resolution imaging (approximately

a few hundred pm before making contact), and soft cantilevers suffer from a ”jump-to-contact”

phenomenon, i.e. when approaching the surface, the cantilever snaps towards the surface ended

by an uncontrolled landing. While true atomic resolution bycontact-mode AFM has been

demonstrated on samples that are chemically inert,16, 17 this method is not feasible for imaging

reactive surfaces where strong attractive short-range forces act. The long-range attractive forces

6

have been compensated in these experiments by pulling at thecantilever (negative loading force)

after jump-to-contact16 or by immersing cantilever and sample in water to reduce the van-der-

Waals attraction.17 Howald et al.18 could partially solve the reactivity problem by passivating

the reactive Si tip with a thin layer of poly-tetra-fluor-ethylen (teflon). The unit cell of Si(111)-

(7×7) was resolved, but atomic resolution was not reported withthat method of tip passivation.

In summary, AFM shares the challenges that are already knownfrom STM and uses many of

its design features (actuators, vibration isolation etc.), but nature has posed four extra problems

for atomic resolution AFM: 1. Jump-to-contact, 2. Non-monotonic short range forces, 3. Strong

long-range background forces and 4. Instrumental noise in force measurements.

3 Frequency modulation atomic force microscopy

Dynamic AFM modes19–21 help to alleviate two of the four major AFM challenges. Jump-to-

contact can be prevented by oscillating the cantilever at a large enough amplitudeA such that

the withdrawing force on the cantilever given byk × A is larger than the maximal attractive

force.22 Because the noise in cantilever deflection measurements hasa component that varies in

intensity inversely with frequency (1/f -noise), dynamic AFM modes are less subject to noise

than quasistatic operating modes. Non-monotonic interactions and strong long-range contribu-

tions are still present.

In amplitude modulation AFM,19 the cantilever is driven at a constant frequency and the

vibration amplitude is a measure of the tip-sample interaction. In 1991, Albrecht et al. have

shown that frequency modulation (FM) AFM20 offers even less noise at larger bandwidth than

amplitude modulation AFM. In FM-AFM, a cantilever with a high quality factorQ is driven

to oscillate at its eigenfrequency by positive feedback with an electronic circuit that keeps the

amplitudeA constant. A cantilever with a stiffness ofk and effective massm has an eigen-

frequency given byf0 = 1/(2π)√

k/m. When the cantilever is exposed to a tip-sample force

7

gradientkts, its frequency changes instantly tof = f0 +∆f = 1/(2π)√

k′/m with k′ = k+kts

(see Fig. 2). Whenkts is small compared tok, the square root can be expanded and the fre-

quency shift is simply given by20

∆f(z) =f0

2kkts(z). (1)

This formula is only correct ifkts is constant over the distance range fromz − A to z + A that

is covered by the oscillating cantilever. The force gradient kts was probably almost constant

within the oscillation interval in the first application of FM-AFM in magnetic force microscopy

by Albrecht et al.,20 where magnetic recording media with magnetic transitions spaced by about

2µm were imaged with a cantilever with a stiffness of about 10 N/m oscillating at an amplitude

of about 5 nm. In contrast, in the newer application of FM-AFMin atomic-resolution AFM, the

force gradient varies by orders of magnitude throughout theoscillation of the cantilever. Using

frequency modulation AFM, true atomic resolution on Si(111)-(7×7), a fairly reactive sample,

could be achieved in 1994.23 Figure 3 shows the topographic image of this data. The fast scan-

ning direction was horizontal, and the atomic contrast is rather poor in the lower section, quite

good in a narrow strip in the center part and vanishing in the top section. These changes in

contrast were due to tip changes, indicating fairly strong interaction during the imaging pro-

cess. A piezoresistive cantilever made of Si24 as shown in Fig. 4(a) with a stiffness of17 N/m

was used to obtain this image. The amplitude of the cantilever can be freely adjusted by the

operator, and while it was even planned to use the thermally excited amplitude25 (≈ 10 pm),

the empirically determined optimal amplitude values were always around 10 nm – on a similar

order of magnitude as the value ofA = 34 nm used in the data of Fig. 3. The chemical bonding

forces that are responsible for the atomic contrast in imaging Si by AFM have a range on the

order of 100 pm,26 and the amplitude was 340 times as large. The requirement of such a large

amplitude is in stark contrast to intuition. Imagine an atommagnified to a size of an orange

8

with a diameter of 8 cm. The range of the bonding force is then only 4 cm or so, and the front

atom of the cantilever would approach from a distance of 20 m and only in the last few cen-

timeters of its oscillation cycle would feel the attractivebonding forces from the sample atom

next to it. On the other hand, force gradients can be quite large in chemical bonds. According

to the well-known Stillinger-Weber potential,27 a classic model potential the interaction of Si

atoms in the solid and liquid phases, a single bond between two Si atoms has a force gradient of

kts ≈ +170 N/m at the equilibrium distance ofz = 235 pm andkts ≈ −120 N/m when the two

Si atoms are at a distance ofz = 335 pm. Because of the relatively large values of interatomic

force gradients, even cantilevers with a stiffness on the order of 1 kN/m should be subject to

significant frequency shifts when oscillating at small amplitudes (page 5 in Ref28).

Nevertheless, the large-amplitude FM technique has celebrated great successes by imaging

metals, semiconductors and insulators with true atomic resolution.29–33

4 The search for optimal imaging parameters

In order to understand why these large oscillation amplitudes were necessary, a quantitative

analysis of the physics of large amplitude FM-AFM was necessary, starting with a calculation

of frequency shift for large amplitudes. Ifkts is not constant over one oscillation cycle, Eq. 1

no longer holds and perturbation theory such as the Hamilton-Jacobi theory34 can be used to

find the relationship between frequency and tip sample forces.22 Other perturbative approaches

have confirmed the result,35–38and an instructive representation of the formula is

∆f(z) =f0

πk

∫ 1

−1kts(z − uA)

√1 − u2du. (2)

This equation is key to a physical understanding of FM-AFM allowing to evaluate the impact

of various force components on∆f , the experimental observable. On a first glance, the large-

amplitude result resembles Eq. 1 wherekts(z) is replaced by an averaged value. The average

9

force gradient is computed by convolutingkts(z) in the intervalz − A to z + A with a semi-

spherical weight function. The weight function has its maximum atu = 0 – a distanceA away

from the minimal tip-sample distance. The minimal tip sample distancezmin is an important

parameter in any STM or AFM experiment, because while a smallvalue ofzmin is desirable

for optimal spatial resolution, both tip and sample can be damaged ifzmin is too small. We

can now ask, if we keepzmin constant and varyA, what happens to our signal, the frequency

shift ∆f? The answer is given in Eq. 2: as long as the gradient of the tip-sample interaction

kts remains constant as the tip of the cantilever moves over az-range fromzmin to zmin + 2A,

∆f stays constant. However, asA reaches the decay lengthλ of the interaction, the frequency

shift drops sharply at a rate∝ (λ/A)3/2. It turns out that for amplitudes larger thanλ, ∆f

is no longer proportional to the force gradient, but to the product of force and the square root

of λ39 (or, equivalently to the geometric average between potential and force40). In FM-AFM

with amplitudes large compared to the interaction range, itis useful to define a quantityγ =

∆fkA3/2/f0.22 The ’normalized frequency shift’γ connects the physical observable∆f and

the underlying forcesFts with rangeλ, whereγ ≈ 0.4Ftsλ1/2 (see Eqs. 35-41 in33). For

covalent bonds, the typical bonding strength is on the orderof -1 nN with λ ≈ 1 A, resulting

in γ ≈ −4 fN√

m, where a negative sign indicates attractive interaction.The crossover from

the small-amplitude approximation in Eq. 1 to the large-amplitude case in Eq. 2 occurs for

amplitudes on the order of the interaction rangeλ.

Equation 2 determines the influence of the oscillation amplitude on AFM challenge number

3 (disturbing contribution of long-range forces) outlinedin the second section: Imagine an

AFM tip at a minimal distancezmin = 0.3 nm to a surface, where the total tip-sample force is

composed of a chemical bonding force with an exponential distance dependence and a given

range with a long-range force with the same strength and a tentimes longer range (see caption

of Table 1 for details). In large-amplitude AFM (here,A > 1 nm), the signal is proportional to

10

the normalized frequency shiftγ, and the long-range contribution to∆f is√

1 nm/100 pm≈ 3-

times larger than the short-range contribution. For small amplitudes (here,A < 100 pm),∆f is

proportional to the force gradient and the long-range component is only100 pm/1 nm = 1/10

of the short-range contribution. Therefore, small amplitude AFM helps to reduce the unwanted

contribution of long-range forces.

Even stronger attenuation of the unwanted long-range contribution would be possible if

higher order force derivatives could be mapped directly. For example, if we could directly mea-

sure∂2Fts/∂z2, the long range component was only 1/100 of the short-range contribution, and

for a direct mapping of the third order gradient∂3Fts/∂z3, the relative long range component

would reduce to a mere 1/1000. Higher force gradients can be mapped directly by higher-

harmonic AFM, as shown further below.

AFM physical short-range long-range relative short-method observable contribution contribution range contribution

quasistatic force 1 nN 1 nN 50 %large amplitude FM γ ≈ 0.4×force×

√range 4 fN

√m 12 fN

√m 25 %

small amplitude FM force gradient 10 N/m 1 N/m 91 %higher-harmonic n-th force gradient 10n+9(n−1) N/mn 109(n−1) N/mn ≈ 100 %(1 − 10−n)

Table 1: Short- and long-range contributions to AFM signalsin different operating modes.This model calculation assumes a chemical bonding forceF (z) = F0e

−z/λ with a strength ofFshort range(zmin) = 1 nN and a range ofλshort range = 100 pm and an equally strong long-rangebackground force withFlong range(zmin) = 1 nN and a range ofλlong range = 1 nm. Dependingon the mode of AFM operation, the short-range part has a different weight in the total interactionsignal. Higher-harmonic AFM offers the greatest attenuation of long-range forces.

Because the forces that act in AFM are small, optimizing the signal-to-noise ratio is crucial

for obtaining good images. Frequency noise in FM-AFM is inversely proportional to ampli-

tude.19, 20, 33, 41 As discussed above, the signal stays constant untilA reachesλ and drops pro-

portional to(λ/A)3/2 for larger amplitudes. Therefore, the signal-to-noise ratio is maximal for

amplitudes on the order of the decay length of the interaction that is used for imaging.42 For

11

atomic imaging, amplitudes on the order of 100 pm are expected to be optimal.

As a conclusion of these calculations, we find that the use of small amplitudesA ≈ λ would

have two advantages:

1. Increased signal-to-noise ratio42

2. Greater sensitivity to short-range forces.33

So why was it not feasible to use small amplitudes in the initial experiments? Two reasons,

related to the mechanical stability of the oscillating cantilever, can be identified. First, jump-to-

contact is prevented if the withdrawing force of the cantilever when it is closest to the sample

given byk × A is larger than the maximal attraction.22 Second, because tip-sample forces are

not conservative,43 random dissipative phenomena with a magnitude ofδEts cause amplitude

fluctuationsδA = δEts/(kA).42, 44 Both problems can be resolved by utilizing cantilevers with

sufficient stiffness. Stability considerations propose a lower threshold fork that depends on the

tip-sample dissipation as well as theQ-factor of the cantilever. Because the frequency shift is

inversely proportional to the stiffness (Eqs. 1 and 2),k should still be chosen as low as permitted

by the stability requirements. Stiff cantilevers were not commercially available when we real-

ized their potential advantages, therefore we built cantilevers with a stiffness ofk = 1800 N/m

from quartz tuning forks45, 46(see Fig. 4(b)). A secondary advantage of quartz cantilevers is their

greater frequency stability with temperature, which leadsto lower frequency drift in particular

if a quartz stabilized frequency detector is used (we used the EasyPLL by Nanosurf AG, Liestal,

Switzerland). Other small-amplitude approaches with stiff home-built tungsten cantilevers have

been demonstrated in Ragnar Erlandsson’s47 and John Pethica’s groups.48–50 As predicted by

theoretical considerations, the stiff cantilever allowedto use sub-nm amplitudes, resulting in an

improved signal-to-noise ratio, a strong attenuation of the disturbing long-range forces and the

possibility of stable scanning at very small tip-sample distances. For these reasons, the spatial

12

resolution was increased as shown in Fig. 5. The image shows avery clear picture of Si with

a defect and very large corrugation. The adatoms of Si which should be spherically symmetric

showed subatomic details that are interpreted as orbitals in the tip atom.51, 52 This AFM image

seemed to show greater resolution than what was known from STM. According to the ‘Stoll-

formula’,53 a theoretical estimate of the vertical corrugation and thusthe lateral resolution of

STM images, two physical parameters are crucial for the highspatial resolution of STM: a)

the very short decay length of the tunneling current and b) a small tip-sample distance. Three

likely reasons have been identified that may explain why dynamic AFM could provide better

resolution than STM:54

1. In dynamic AFM, the minimal tip sample distance can be muchsmaller than in STM

without destroying the tip, because the shear forces that act on the front atom during

scanning are much smaller in the oscillation phase where thetip is far from the sample.

2. When using large gap voltages, a variety of states can contribute to the tunneling current,

smearing out the image.

3. Tip-sample forces also have repulsive components with a very short decay length.

The first two characteristics can be fulfilled in STM as well byusing a very small tunneling bias

voltage and oscillating the STM tip similar to an AFM tip. Figure 6 shows an image obtained

in dynamic STM where a Co6Fe3Sm magnetic tip was mounted onto a qPlus sensor, imaging

Si.55, 56 Each Si adatom looks like a fried egg with a sharp center peak surrounded by a halo.

The radius of the center peak is only on the order of 100 pm, showing that higher-momentum

states57 must have been involved in this image. The experiment was repeated with pure Co, Fe

and Sm tips, and only pure Sm tips yielded similar images as Figure 6, we therefore concluded

that a Sm atom acted as the tip atom in this experiment.55 In atomic samarium, the electrons at

highest occupied state are in a 4f state. If one assumes, thatthe electronic states at a Sm surface

13

atom of bulk Co6Fe3Sm are similar to atomic states in Sm, it appears likely that the crystal field

around the front atom creates a state close to 4fz3 symmetry that is responsible for the tunneling

contrast. Interestingly, very small tip-sample distancescould only be realized with oscillating

tips. When the oscillation was turned off, the current setpoint had to be reduced otherwise the

tip would not survive the small tunneling distances.

Operation at small oscillation amplitudes not only resultsin greater resolution, it also facil-

itates simultaneous STM and AFM imaging. A straightforwardimplementation of combined

current- and force measurements uses the constant-height mode, where thez-position of the tip

is held constant relative to the plane connecting the surface atoms. A simultaneous measurement

of tunneling current and frequency shift allows to compare forces and tunneling currents. Figure

7 shows a comparison of current and repulsive force on graphite58 observed by simultaneous

AFM and STM in vacuum at liquid helium temperatures (4.9 K). STM only sees the electrons

at the Fermi level, while repulsive forces act wherever the local charge density is high (i.e. over

everyatom) for small enough distances. In graphite, only every second surface atom conducts

electricity, but every surface atom exerts repulsive forces. Therefore, AFM “sees more” than

STM and allows to correlate topography to local conductance. This method is promising for

other materials with more than one basis atom in the elementary cell. The images have been

taken with a low-temperature AFM/STM operating at 4.9 K in ultra-high vacuum.59, 60

While a strong bias dependence holds both for atomic-resolution STM61 as well as AFM

images,60, 62 one pronounced difference is that the direction of the tunneling current is not ac-

cessible in STM, while the direction of the measured force isdetermined by the orientation of

the cantilever. Usually, AFM senses forces that are normal to the surface, but it is also possible

to perform lateral force microscopy63 by measuring the forces acting parallel to the surface. In a

quasistatic mode, lateral forces can be recorded simultaneously with normal forces. In dynamic

modes, it is easier to rotate the attachment of the cantilever by 90 degrees and detect lateral

14

forces. Figure 8 shows a measurement of the lateral force gradients between a tip and a Si

surface. Parallel motion between tip and cantilever also allows to use extremely soft cantilevers

without suffering jump-to-contact to probe the limits of force resolution, as demonstrated by

Rugar et al. in single spin detection by magnetic resonance force microscopy.64

5 Higher-harmonic atomic force microscopy

Can we increase the spatial resolution of AFM any further? When decreasing the amplitude

from A >> λ to A << λ, the frequency shift changes from a proportionality ofFts

√λ to

Fts/λ. As outlined above, an experimental observable that is proportional to a higher force

gradient should allow even higher spatial resolution than small-amplitude FM-AFM. Luckily,

there is a physical observable that couples directly to higher force gradients. When the can-

tilever oscillates in the force field of the sample, a shift infrequency is not the only change

in the cantilever’s motions. The oscillation of the cantilever changes from a purely sinu-

soidal motion given byq′ = A cos(2πft) to an oscillation that contains higher harmonics with

q′ =∑

∞

n=0 an cos(2πnft + φn). For amplitudes that are large with respect to the range of

Fts, the higher harmonics are essentially proportional to∆f .37 However, for small amplitudes,

Durig has found thatFts could be recovered immediately within the distance range from zmin

to zmin + 2A if the amplitudes and phases of all higher harmonics of the cantilever’s motion

were known.65 Moreover, higher harmonics bear even more useful information: direct cou-

pling to higher force gradients.66 Similar to Eq. 2, we can express the magnitude of the higher

harmonics by a weighted average of a force gradient – a gradient of ordern > 1 this time:

an =2

πk

1

1 − n2

An

1 · 3 · ... · (2n − 1)

∫ 1

−1

dnFts(z + Au)

dzn(1 − u2)n−1/2du. (3)

The weight function changes from the semi-spherical shapew∆f(u) = (1 − u2)1/2 in Eq. 2 to

functionswn(u) = (1 − u2)n−1/2 that are more and more peaked with increasingn. For this

15

reason, the use of small amplitudes is of even greater importance in higher-harmonic AFM than

in FM-AFM. The magnitude of the higher harmonic amplitudesan is rather small compared

to the fundamental amplitudea1 = A, therefore higher harmonic AFM works best at low

temperatures, where the detection bandwidth can be set to very small values.

The spatial resolution of AFM and STM is fundamentally neither limited by the mechanical

vibration level nor by thermal vibrations, but by the spatial extent of the experimental objects

that are observed – electrons at the Fermi level in STM,67 and something close to the total charge

density in repulsive AFM.68 When probing the resolution limits of AFM, we first have to find

an object with the desired sharply localized electronic states. Pauling69 has noted, that transi-

tion metals show a covalent bonding character, and should therefore expose lobes of increased

charge density towards their neighbors. Indeed, while the surface atoms of W(001) expose a

large blurred charge cloud at the Fermi level fork-vectors perpendicular to the surface (Fig. 8

in Ref.70), the total charge density shows four distinct maxima (Fig.3 in Ref.70 and Fig. 3(a) in

Ref.71). Figure 9 shows a direct comparison of the simultaneously recorded tunneling current

and the amplitudes of the higher harmonics. As expected, thehigher harmonic data shows much

greater detail.

6 Summary and Conclusion

We have emphasized the enormous usefulness of AFM by referring to the numerous references

to the original publication2 in the introduction. While most AFM applications are currently not

in the atomic resolution regime, the enhancement in spatialresolution is likely to add significant

value in most AFM studies in physics, chemistry, biology andmaterials science. Recently, true

atomic resolution by FM-AFM has been observed at ambient pressure in an N2 atmosphere,72

showing that some of the concepts of vacuum AFM are applicable in ambient environments.

Although STM resolution can benefit from oscillating the tip, a concept that has originated in

16

AFM, Fig. 9 shows that AFM has now clearly reached and even surpassed the resolution ca-

pability of STM. Figure 10 shows the evolution of the resolution of AFM from large amplitude

AFM in 1994 (a) to small amplitude AFM in 2000 (b) and higher harmonic AFM in 2004 (c).

While the structures within single atoms shown in Fig. 10 (b)and (c) originate in the front

atom of the probe, other examples where AFM shows more atomicdetails of specimens than

STM such as the observation of the rest atoms in Si(111)-(7×7)73, 74 or the observation of all

dangling bonds on the Si/Ge(105) surface75 establish the improved spatial resolution of AFM

over STM in special cases. Atomic- and molecular structuring has been the domain of STM for

a long time, starting with the first demonstration of manipulating single atoms76 to a variety of

nano-structuring methods by STM.77 Recently, it has been shown that atomic manipulation by

AFM is possible even at room temperature.78

We have not been able to discuss the phenomenal success of AFMin biology, a field with

a much more immediate impact on the human condition. It can beexpected that at least some

of the concepts that have been developed for AFM in vacuum will enable greater resolution in

biological AFM applications as well.79, 80

Acknowledgments

I wish to thank Jochen Mannhart for support and editorial suggestions and my current

and former students Martin Breitschaft, Philipp Feldpausch, Stefan Hembacher, Markus Herz,

Christian Schiller, Ulrich Mair, Thomas Ottenthal and Martina Schmid for their contributions

towards the progress of AFM. I also thank Gerd Binnig, CalvinF. Quate and Christoph Ger-

ber for kicking off the fun of the AFM field and for ongoing inspiring interactions. Special

thanks to Heinrich Rohrer, Calvin Quate and Christoph Gerber for critical comments and to

German Hammerl for help with LaTeX. Supported by the Bundesministerium fur Forschung

und Technologie under contract 13N6918.

17

References

1. Binnig, G.Atomic Force Microscope and Method for Imaging Surfaces with Atomic Reso-

lution (1986)US Patent 4,724,318

2. Binnig, G., Quate, C. F., and Gerber, C.Phys. Rev. Lett.(1986)56, 930

3. Binnig, G. and Rohrer, H.Rev. Mod. Phys.(1999)71, S324

4. Faraday, M.Chemical History of a Candle. reprinted in 2003, Dover Publications, Mineola,

NY (1861)

5. Siegman, A. E.Lasers. University Science Books, Herndon, VA, USA (1986)

6. Binnig, G., Rohrer, H., Gerber, C., and Weibel, E.Phys. Rev. Lett.(1982)49, 57

7. Hofer, W. A.Materials Today(2002)October, 24

8. Binnig, G., Rohrer, H., Gerber, C., and Weibel, E.Phys. Rev. Lett.(1983)50, 120

9. Takayanagi, K., Tanishiro, Y., Takahashi, M., and Takahashi, S. J. Vac. Sci. Technol. A

(1985)3, 1502

10. Binnig, G., Gerber, C., Stoll, E., Albrecht, T. R., and Quate, C. F.Europhys. Lett.(1987)3,

1281

11. Albrecht, T. R. and Quate, C. F.J. Appl. Phys.(1987)62, 2599

12. Meyer, G. and Amer, N. M.Appl. Phys. Lett.(1990)56, 2100

13. Meyer, E., Heinzelmann, H., Rudin, H., and Guntherodt,H.-J. Z. Phys. B(1990)79, 3

14. Riordon, J.APS News(2003)May issue, 3

18

15. Rugar, D. and Hansma, P.Physics Today(1990)43, 23

16. Giessibl, F. J. and Binnig, G.Ultramicroscopy(1992)42-44, 281

17. Ohnesorge, F. and Binnig, G.Science(1993)260, 1451

18. Howald, L., Luthi, R., Meyer, E., and Guntherodt, H.-J. Phys. Rev. B(1995)51, 5484

19. Martin, Y., Williams, C. C., and Wickramasinghe, H. K.J. Appl. Phys.(1987)61, 4723

20. Albrecht, T. R., Grutter, P., Horne, H. K., and Rugar, D.J. Appl. Phys.(1991)69, 668

21. Durig, U., Zuger, O., and Stalder, A.Journal of Applied Physics(1992)72, 1778

22. Giessibl, F. J.Phys. Rev. B(1997)56, 16010

23. Giessibl, F. J.Science(1995)267, 68

24. Tortonese, M., Barrett, R. C., and Quate, C.Appl. Phys. Lett.(1993)62, 834

25. Giessibl, F. J.Jpn. J. Appl. Phys.(1994)33, 3726

26. Perez, R., Stich, I., Payne, M. C., and Terakura, K.Phys. Rev. Lett.(1997)78, 678

27. Stillinger, F. H. and Weber, T. A.Phys. Rev. B(1985)31, 5262

28. Giessibl, F. J.German Patent(1996) Patentschrift DE 196 33 546

29. Morita, S., Wiesendanger, R., and Meyer, E., eds.Noncontact Atomic Force Microscopy.

Nanoscience and Technology. Springer Berlin Heidelberg New York (2002)

30. Meyer, E., Hug, H. J., and Bennewitz, R.Scanning Probe Microscopy. The Lab on a Tip.

Springer, Berlin Heidelberg New York (2003)

31. Garcia, R. and Perez, R.Surf. Sci. Rep.(2002)47, 197

19

32. Hofer, W. A., Foster, A. S., and Shluger, A. L.Rev. Mod. Phys.(2003)75, 1287

33. Giessibl, F. J.Rev. Mod. Phys.(2003)75, 949

34. Goldstein, H.Classical Mechanics. Addison Wesley, Reading MA (1980)

35. Baratoff, A.unpublished(1997)

36. Durig, U.Surf. and Interf. Anal.(1999)27, 467

37. Durig, U.Appl. Phys. Lett.(1999)75, 433

38. Livshits, A., Shluger, A., and Rohl, A.Appl. Surf. Sci.(1999)140, 327

39. Giessibl, F. J. and Bielefeldt, H.Phys. Rev. B(2000)61, 9968

40. Ke, S. H., Uda, T., and Terakura, K.Phys. Rev. B(1999)59, 13267

41. Hasegawa, Y., Eguchi, T., An, T.,et al.. Japanese Journal of Applied Physics(2004)43,

L303

42. Giessibl, F. J., Bielefeldt, H., Hembacher, S., and Mannhart, J.Appl. Surf. Sci.(1999)140,

352

43. Kantorovich, L. N.Phys. Rev. B(2001)64, 245409

44. Giessibl, F. J., Hembacher, S., Herz, M., Schiller, C., and Mannhart, J.Nanotechnology

(2004)15, S79

45. Giessibl, F. J.Appl. Phys. Lett.(1998)73, 3956

46. Giessibl, F. J.Appl. Phys. Lett.(2000)76, 1470

47. Erlandsson, R., Olsson, L., and Martensson, P.Phys. Rev. B(1997)54, R8309

20

48. Hoffmann, P. M., Oral, A., Grimble, R. A.,et al.. Proc. R. Soc. London A(2001)457, 1161

49. Hoffmann, P. M., Jeffery, S., Pethica, J. B.,OzgurOzer, H., and Oral, A.Physical Review

Letters(2001)87, 265502

50. Oral, A., Grimble, R. A.,OzgurOzer, H., Hoffmann, P. M., and Pethica, J. B.Appl. Phys.

Lett. (2001)79, 1915

51. Giessibl, F. J., Hembacher, S., Bielefeldt, H., and Mannhart, J.Science(2000)289, 422

52. Huang, M., Cuma, M., and Liu, F.Phys. Rev. Lett.(2003)90, 256101

53. Stoll, E.Surf. Sci. Lett.(1984)143, L411

54. Giessibl, F. J., Bielefeldt, H., Hembacher, S., and Mannhart, J.Ann. Phys. (Leipzig)(2001)

10, 887

55. Herz, M., Giessibl, F. J., and Mannhart, J.Phys. Rev. B(2003)68, 045301

56. Herz, M. Dynamische Tunnel-, Kraft- und Reibungsmikroskopie mit atomarer und sub-

atomarer Auflosung. Ph.D. thesis, University of Augsburg, Germany (2003)

57. Chen, C. J.Introduction to Scanning Tunneling Microscopy. Oxford University Press, New

York (1993)

58. Hembacher, S., Giessibl, F. J., Mannhart, J., and Quate,C. F. Proc. Natl. Acad. Sci. (USA)

(2003)100, 12539

59. Hembacher, S. F.Simultane Rasterkraft- und Rastertunnelmikroskopie bei 5K im Ultra-

hochvakuum. Ph.D. thesis, Universitat Augsburg, Germany (2003)

60. Hembacher, S., Giessibl, F. J., and Mannhart, J.Phys. Rev. Lett.(2005)94, 056101

21

61. Feenstra, R. M., Stroscio, J. A., Tersoff, J., and Fein, A. P. Phys. Rev. Lett.(1987)58, 1192

62. Arai, T. and Tomitori, M.Phys. Rev. Lett.(2004)93, 256101

63. Mate, M., McClelland, G. M., Erlandsson, R., and Chiang,C. Phys. Rev. Lett.(1987)59,

1942

64. Rugar, D., Budakian, R., Mamin, H. J., and Chui, B.Nature(2004)430, 329

65. Durig, U.New Journal of Physics(2000)2, 5.1

66. Hembacher, S., Giessibl, F. J., and Mannhart, J.Science(2004)305, 380

67. Tersoff, J. and Hamann, D. R.Phys. Rev. Lett.(1985)50, 1998

68. Ciraci, S., Baratoff, A., and Batra, I. P.Phys. Rev. B(1990)41, 2763

69. Pauling, L. The Nature of the Chemical Bond. Cornell Univ. Press, Ithaca, New York

(1957)

70. Posternak, M., Krakauer, H., Freeman, A. J., and Koelling, D. D. Phys. Rev. B(1980)21,

5601

71. Mattheiss, L. F. and Hamann, D. R.Phys. Rev. B(1984)29, 5372

72. Sasahara, A., Kitamura, S., Uetsuka, H., and Onishi, H.Journal of Physical Chemistry B

(2004)108, 15735

73. Lantz, M., Hug, H. J., van Schendel, P.,et al.. Phys. Rev. Lett.(2000)84, 2642

74. Eguchi, T. and Hasegawa, Y.Phys. Rev. Lett.(2002)89, 266105

75. Eguchi, T., Fujikawa, Y., Akiyama, K.,et al.. Phys. Rev. Lett.(2004)93, 266102

22

76. Eigler, D. M. and Schweizer, E. K.Nature(1990)344, 524

77. Rosei, F.Journal of Physics-Condensed Matter(2004)16, S1373

78. Sugimoto, Y., Abe, M., Hirayama, S.,et al.. Nature Materials(2005)4, 156159

79. Miles, M. Science(1997)277, 1845

80. Horber, J. K. H. and Miles, M. J.Science(2003)302, 1002

23

Figures

Figure 1: (a) Schematic view of a tip and a sample in a scanningtunneling microscope oratomic force microscope. The diameter of a metal atom is typically 0.3 nm. (b) Qualitativedistance dependence of tunneling current, long- and short-range forces. The tunneling currentincreases monotonic with decreasing distance, while the force reaches a minimum and increasesfor distances below the bond length.

24

Figure 2: (a) Schematic view of a vibrating tip close to a a sample in a dynamic atomic forcemicroscope. The forces that act between the tip and the sampleFts cause a detectable change inthe oscillation properties of the cantilever. (b) Mechanical equivalent of (a). The free cantileverwith stiffnessk and effective massm can be treated as a harmonic oscillator with an eigen-frequencyf0 = (k/m)1/2/(2π). The bond between tip and sample with its stiffnesskts altersthe resonance frequency tof = ([k + kts]/m)1/2/(2π). When the oscillation amplitude of thecantilever is large,kts can vary significantly within one oscillation cycle, requiring averaging(see text).

25

Figure 3: First AFM image of a reactive surface showing true atomic resolution: Si(111)-(7×7)reconstruction. Parameters:k = 17 N/m, A = 34 nm,f0 = 114 kHz, ∆f = −70 Hz andQ =28 000, scanning speed = 3.2 lines/s. Environment: ultra-high vacuum, room temperature.23

26

Figure 4: Micrographs of (a) a piezoresistive cantilever24 and (b) a ‘qPlus’ sensor46 - a cantilevermade from a quartz tuning fork. The piezoresistive cantilever has a length of 250µm, a widthof 50µm and a thickness of 4µm. The eigenfrequency is 114 kHz, the stiffness 17 N/m and theQ-factor in vacuum 28 000. The qPlus sensor has a typical eigenfrequency ranging from 10 to30 kHz (depending on the mass of the tip), a stiffness of 1 800 N/m and aQ-factor of 4000 invacuum atT = 300 K and 20 000 atT = 4 K. One of the prongs is fixed to a large substrateand a tip is mounted to the free prong. Because the fixed prong is attached to a heavy mass, thedevice is mechanically equivalent to a traditional cantilever. The dimensions of the free prongare: Length: 2.4 mm, width: 130µm, thickness: 214µm.

27

Figure 5: AFM image of the silicon 7×7 reconstruction with true atomic resolution with a stiffcantilever. Parameters:k = 1800 N/m, A = 0.8 nm, f0 = 16.86 kHz, ∆f = −160 Hz andQ = 4 000. Environment: ultra-high vacuum, room temperature.51

28

Figure 6: (a) Dynamic STM image of the silicon 7×7 reconstruction where a Co6Fe3Sm tip wasmounted on a qPlus sensor. Parameters:k = 1800 N/m, A = 0.5 nm,f0 = 19 621Hz, samplebias voltage -100 mV, average tunneling current 200 pA. (b) Schematic plot of tip and samplestates that can lead to the experimental image shown in (a). The sample state is a dangling bondof a Si adatom with its 3sp3 symmetry, while a Sm 4fz3 state is taken as a tip state. Environment:ultra-high vacuum, room temperature.56

29

Figure 7: (a) Constant height STM image of graphite and (b) a simultaneously recorded AFMimage (repulsive). Part (c) shows an estimate of the charge density at the Fermi level (visibleby STM) and (d) the total charge density (relevant for repulsive AFM) for graphite. Parameters:k = 1800N/m, A = 0.3 nm,f0 = 18076.5 Hz, andQ = 20 000.58

30

Figure 8: (a) Topographic STM image of Si(111)-(7×7) where the tip is mounted on a lateralforce sensor. The tip oscillates withA ≈ 80 pm in they-direction in the lower half of theimage, the oscillation is turned off in the upper half. (b) Corresponding lateral force gradient.On top of the adatoms, the bond between tip and sample causes an increase in frequency shift.Parameters:k = 1350 N/m, A = 80 pm (bottom),A = 0 (top),f0 = 10214 Hz. Environment:ultra-high vacuum, room temperature.56

31

Figure 9: Simultaneous constant height STM (left column) and higher-harmonic AFM images(center column) of graphite with a tungsten tip. The right column shows the proposed orienta-tion of the W tip atom. The W atom is represented by its Wigner-Seitz unit cell, which reflectsthe full symmetry of the bulk. We assume, that the bonding symmetry of the adatom is similarto the bonding symmetry of the bulk. This assumption is basedon charge density calculationsof surface atoms70, 71 In the first row, the higher harmonics show a two-fold symmetry, as result-ing from a [110] orientation of the front atom. In the second row, the higher harmonics showroughly a three-fold symmetry, as expected for a [111] orientation. In the third row, the sym-metry of the higher-harmonic signal is approximately four-fold, as expected for a tip in [001]orientation. Parameters:k = 1800N/m, A = 0.3 nm, f0 = 18 076.5 Hz, andQ = 20 000.Environment: ultra-high vacuum,T = 4.9 K.66

32

Figure 10: Progress in spatial resolution of AFM showing images of single atoms. The lateralscale in (a)-(c) is equal. (a) An adatom of the Si(111)-(7×7) reconstruction, showing up asa blurred spot. (b) An adatom of the Si(111)-(7×7) reconstruction, showing subatomic con-trast originating in the electronic structure of the tip. (c) Higher-harmonic image of a tungstenatom mapped by a carbon atom. Parameters: (a)k = 17 N/m, A = 34 nm, f0 = 114 kHz,∆f = −70 Hz andQ = 28 000 (ultra-high vacuum, room temperature), (b)k = 1800 N/m,A = 0.8 nm, f0 = 16 860Hz, ∆f = −160 Hz andQ = 4 000 (ultra-high vacuum, room tem-perature), (c)k = 1800N/m, A = 0.3 nm, f0 = 18 076.5 Hz, andQ = 20 000 (ultra-highvacuum,T = 4.9 K), higher harmonic detection. (d) Schematic view of a Si(001) tip close to aSi(111)-(7×7) surface. Because of the large amplitude and a fairly largeminimum tip-sampledistance, the blurry image (a) corresponding to this configuration is approximately symmetricwith respect to the vertical axis. (e) Similar to (d), but at acloser distance. The angular depen-dence of the bonding forces is noticeable. (f) W(001) surface close to a C atom in a graphitesurface. The charge distribution in W shows small pockets that are resolved by higher-harmonicAFM with a light-atom carbon-probe.

33